A circle has a radius of $8$. An arc in this circle has a central angle of $\dfrac{5}{3}\pi$ radians. What is the length of the arc? ${16\pi}$ ${\dfrac{5}{3}\pi}$ $\color{#DF0030}{\dfrac{40}{3}\pi}$ ${8}$
Answer: First, calculate the circumference of the circle. $c = 2\pi r = 2\pi (8) = 16\pi$ The ratio between the arc's central angle $\theta$ and $2 \pi$ radians is equal to the the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{2 \pi} = \dfrac{s}{c}$ $\dfrac{5}{3}\pi \div 2 \pi = \dfrac{s}{16\pi}$ $\dfrac{5}{6} = \dfrac{s}{16\pi}$ $\dfrac{5}{6} \times 16\pi = s$ $\dfrac{40}{3}\pi = s$